\(\int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx\) [260]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\text {Int}\left (\frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x},x\right ) \]

[Out]

CannotIntegrate(arccoth(1+I*d+d*cot(b*x+a))/x,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \]

[In]

Int[ArcCoth[1 + I*d + d*Cot[a + b*x]]/x,x]

[Out]

Defer[Int][ArcCoth[1 + I*d + d*Cot[a + b*x]]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \]

[In]

Integrate[ArcCoth[1 + I*d + d*Cot[a + b*x]]/x,x]

[Out]

Integrate[ArcCoth[1 + I*d + d*Cot[a + b*x]]/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\operatorname {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right )}{x}d x\]

[In]

int(arccoth(1+I*d+d*cot(b*x+a))/x,x)

[Out]

int(arccoth(1+I*d+d*cot(b*x+a))/x,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]

[In]

integrate(arccoth(1+I*d+d*cot(b*x+a))/x,x, algorithm="fricas")

[Out]

integral(1/2*log(((d - I)*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/d)/x, x)

Sympy [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\operatorname {acoth}{\left (d \cot {\left (a + b x \right )} + i d + 1 \right )}}{x}\, dx \]

[In]

integrate(acoth(1+I*d+d*cot(b*x+a))/x,x)

[Out]

Integral(acoth(d*cot(a + b*x) + I*d + 1)/x, x)

Maxima [N/A]

Not integrable

Time = 5.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 7.05 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]

[In]

integrate(arccoth(1+I*d+d*cot(b*x+a))/x,x, algorithm="maxima")

[Out]

-I*b*x + 1/4*(-I*pi - 4*I*a - 2*log(d))*log(x) + 1/2*I*integrate(arctan2(d*cos(2*b*x + 2*a) + sin(2*b*x + 2*a)
, -d*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) - 1)/x, x) + 1/4*integrate(log((d^2 + 1)*cos(2*b*x + 2*a)^2 + (d^2 +
1)*sin(2*b*x + 2*a)^2 + 2*d*sin(2*b*x + 2*a) - 2*cos(2*b*x + 2*a) + 1)/x, x)

Giac [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]

[In]

integrate(arccoth(1+I*d+d*cot(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(arccoth(d*cot(b*x + a) + I*d + 1)/x, x)

Mupad [N/A]

Not integrable

Time = 5.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\mathrm {acoth}\left (d\,\mathrm {cot}\left (a+b\,x\right )+1+d\,1{}\mathrm {i}\right )}{x} \,d x \]

[In]

int(acoth(d*1i + d*cot(a + b*x) + 1)/x,x)

[Out]

int(acoth(d*1i + d*cot(a + b*x) + 1)/x, x)